Consider a parallel plate capacitor with plates in the shape of square and in \(X Y\)-plane. The gap between the plates is filled with dielectric material. The dielectric constant \(k\) varies with \(X\)-axis as \(k(x)=\left[1+\left(\frac{x}{L}\right)^\alpha\right]\), where \(\alpha\) is a constant. Let \(C_d\) and \(C_a\) are capacitance in the presence of dielectric and air, respectively. If the ratio \(\frac{C_d}{C_a}=\frac{7}{6}\), then the value of \(\alpha\) must be

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.